Victor Andreevich Toponogov (; March 6, 1930 – November 21, 2004) was an outstanding

Russia Russia, or the Russian Federation, is a country spanning Eastern Europe and North Asia. It is the list of countries and dependencies by area, largest country in the world, and extends across Time in Russia, eleven time zones, sharing Borders ...

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...

, noted for his contributions to

differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...

and so-called

Riemannian geometry Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, defined as manifold, smooth manifolds with a ''Riemannian metric'' (an inner product on the tangent space at each point that varies smooth function, smo ...

"in the large".

Biography

After finishing secondary school in 1948, Toponogov entered the department of Mechanics and Mathematics at

Tomsk State University The National Research Tomsk State University, TSU () is a public research university located in Tomsk, Russia. The university, which opened in 1888, was the first university in the Asian part of Russia and, in practice, the first Russian univ ...

, graduated with honours in 1953, and continued as a graduate student there until 1956. He moved to an institution in

Novosibirsk Novosibirsk is the largest city and administrative centre of Novosibirsk Oblast and the Siberian Federal District in Russia. As of the 2021 Russian census, 2021 census, it had a population of 1,633,595, making it the most populous city in Siber ...

in 1956 and lived in that city for the rest of his career. Since the institution at Novosibirsk had not yet been fully credentialed, he had defended his Ph.D. thesis at Moscow State University in 1958, on a subject in

Riemann space In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the n-sphere, hyperbolic space, and smooth surfaces in t ...

Novosibirsk State University Novosibirsk State University (NSU) is a public research university located in Novosibirsk, Russia. The university was founded in 1958, on the principles of integration of education and science, early involvement of students with research act ...

was established in 1959. In 1961, Toponogov became a professor at a newly created Institute of Mathematics and Computing in Novosibirsk affiliated with the state university. Toponogov's scientific interests were influenced by his advisor Abram Fet, who taught at Tomsk and later at Novosibirsk. Fet was a well-recognized topologist and specialist in variational calculus in the large. Toponogov's work was also strongly influenced by the work of

Aleksandr Danilovich Aleksandrov Aleksandr Danilovich Aleksandrov (; 4 August 1912 – 27 July 1999) was a Soviet and Russian mathematician, physicist, philosopher and mountaineer. Personal life Aleksandr Aleksandrov was born in 1912 in Volyn, Ryazan Oblast. His father was ...

. Later, the class of

metric space In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...

s known as CAT(''k'') spaces would be named after

Élie Cartan Élie Joseph Cartan (; 9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups, differential systems (coordinate-free geometric formulation of PDEs), and differential geometry. He ...

, Aleksandrov and Toponogov. Toponogov published over forty papers and some books during his career. His works are concentrated in Riemannian geometry "in the large". A significant number of his students also made notable contributions in this field.

Conjecture on Complete Convex Surfaces

In 1995 Toponogov made the conjecture:

''On a complete convex surface S homeomorphic to a plane the following equality holds:'' ''where $\kappa_1$ and $\kappa_2$ are the principal curvatures of S.''

In words, it states that every complete convex surface homeomorphic to a plane must have an umbilic point which may lie at infinity. As such, it is the natural open analog of the

Carathéodory conjecture In differential geometry, the Carathéodory conjecture is a mathematical conjecture attributed to Constantin Carathéodory by Hans Ludwig Hamburger in a session of the Berlin Mathematical Society in 1924.''Sitzungsberichte der Berliner Mathematisch ...

for closed convex surfaces. In the same paper, Toponogov proved the conjecture under either of two assumptions: the integral of the Gauss curvature is less than

2\pi

, or the Gauss curvature and the gradients of the curvatures are bounded on ''S''. The conjecture was proven by Brendan Guilfoyle and Wilhelm Klingenberg in 2024 for

C^

-smooth surfaces.

References

External links

Toponogov's Biography, including a list of his publications
(in English) 1930 births 2004 deaths 20th-century Russian mathematicians 21st-century Russian mathematicians Moscow State University alumni Tomsk State University alumni People from Tomsk Academic staff of Novosibirsk State University Differential geometers Soviet mathematicians Burials at Yuzhnoye Cemetery (Novosibirsk) {{Russia-mathematician-stub

Biography

Conjecture on Complete Convex Surfaces

See also

References

External links